Finite Group Databasebeta

← #10800427

Subgroups — #10800427

Order 768.

1186 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5578C211yesmaximal, characteristic
2C2×#4707C211yesmaximal, characteristic
3#73753C211yesmaximal, characteristic
4C3×#1763C211yesmaximal, characteristic, complemented
5#71631C211yesmaximal, characteristic, complemented
6#70271C212yesmaximal
7#70225C212yesmaximal, complemented
8#69813C211yesmaximal, characteristic, complemented
9#69475C211yesmaximal, characteristic, complemented
10#68884C212yesmaximal
11#68803C212yesmaximal, complemented
12#21409—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#946C2211yescharacteristic
15C2×#944C2211yescharacteristic
16C22×ΓL(1,𝔽25)C2212yes—
17#5578—24no—
18#5578C418yescomplemented
19C2×#897C2211yescharacteristic
20C2×#896—21no—
21C2×#889—21no—
22C2×#888C2211yescharacteristic
23C2×#872—21no—
24C2×#872—21no—
25C2×C42×D3C2211yescharacteristic
26C2×C4×Dic6—21no—
27C2×C4×Dic6C2211yescharacteristic
28C2×C3×#103C2212yes—
29C2×C3×#97C2211yescharacteristic
30C3×C4×C4⋊C4C2211yescharacteristic
31C2×#835C2212yes—
32C2×#816C2212yes—
33C2×C8×Dic3C2212yes—
34#4987C2212yes—
35#4975C2212yes—
36C2×#810C2212yes—
37C4×C4⋊Dic3C2211yescharacteristic
38C4×Dic3⋊C4—21no—
39C2×#803C2211yescharacteristic
40#4707—24no—
41#4667—21no—
42#4650—21no—
43C3×#340C2212yes—
44#4551C2212yes—
45#4529C2212yes—
46#4529C2212yes—
47C2×#533—31no—
48C2×C4×M16—31no—
49#2285—31no—
50#1990—32no—
51#1956—32no—
52#1894—32no—
53#1894—32no—
54#1763—31no—
55#1763—31no—
56#1763—31no—
57#1763D311yescharacteristic, complemented
58C22×C3×Q8C2311yescharacteristic
59C22×C4×D3D411yescharacteristic
60C22×C4×D3D411yescharacteristic
61C22×C4×D3C2311yescharacteristic
62C22×Dic6—21no—
63C22×Dic6C2311yescharacteristic
64C22×C3×C8C2312yes—
65C3×C4×Q8—22no—
66C3×C4×Q8C2×C414yes—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C42—21no—
71C2×C3×C42—21no—
72C2×C3×C42C2311yescharacteristic
73#946—22no—
74#946C2×C414yes—
75#944—22no—
76#944C2×C414yes—
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—22no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3C2312yes—
82C2×C4⋊Dic3C2311yescharacteristic
83C2×Dic3⋊C4—21no—
84C2×Dic3⋊C4—21no—
85C2×Dic3⋊C4—21no—
86C2×Dic3⋊C4—21no—
87C2×Dic3⋊C4—21no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—22no—
90C2×C4×Dic3—21no—
91C2×C4×Dic3—21no—
92C2×C4×Dic3D411yescharacteristic
93C2×C4×Dic3D411yescharacteristic
94C2×C4×Dic3C2311yescharacteristic
95C22×C3⋊C8C2312yes—
96C2×ΓL(1,𝔽25)—24no—
97C2×ΓL(1,𝔽25)—24no—
98C2×ΓL(1,𝔽25)—24no—
99#897—22no—
100#897C2×C414yes—
101#896—28no—
102#889—28no—
103#888—22no—
104#888C2×C414yes—
105#872—28no—
106#872—44no—
107C42×D3C2×C414yes—
108C42×D3D414yes—
109C4×Dic6—22no—
110C4×Dic6—28no—
111C4×Dic6C2×C414yes—
112C3×#103—24no—
113C3×#97—22no—
114C3×#95—21no—
115#835—24no—
116C3⋊#95—21no—
117C3⋊#95—22no—
118#816—24no—
119C8×Dic3—24no—
120#810—24no—
121#803—22no—
122C22×M16—32no—
123#533—38no—
124#533—64no—
125C2×C4⋊Q8—31no—
126C2×C4⋊Q8—31no—
127C2×#125—61no—
128C2×C22⋊Q8—31no—
129C2×C22⋊Q8—31no—
130C2×C22⋊Q8—61no—
131C2×C4×Q8—31no—
132C2×C4×Q8—61no—
133C2×C4×Q8C2×D311yescharacteristic
134C2×#117—61no—
135C2×#117—61no—
136C22×C42—31no—
137C2×#103—32no—
138C2×#103—32no—
139C2×#103—32no—
140C2×#103C2×D312yes—
141C4×M16—64no—
142C2×#97—31no—
143C2×#97C2×D311yescharacteristic
144C2×C4×C8—32no—
145#384—32no—
146#382—61no—
147#380—32no—
148#380—61no—
149C4×C4⋊C4—31no—
150C4×C4⋊C4—61no—
151C4×C4⋊C4C2×D311yescharacteristic
152#340—32no—
153#340—32no—
154#340—32no—
155#340C2×D312yes—
156C23×D3C2×D411yescharacteristic
157C2×C3×Q8—21no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8C22×C412yes—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2411yescharacteristic
167C22×Dic3—21no—
168C22×Dic3—21no—
169C22×Dic3—22no—
170C22×Dic3C2×D411yescharacteristic
171C22×Dic3C2×D411yescharacteristic
172C22×Dic3C2×D411yescharacteristic
173C2×C4×D3—22no—
174C2×C4×D3—22no—
175C2×C4×D3—22no—
176C2×C4×D3—22no—
177C2×C4×D3—24no—
178C2×C4×D3C22⋊C414yes—
179C2×C4×D3C22⋊C414yes—
180C2×C4×D3C22×C412yes—
181C2×C4×D3C2×D412yes—
182C2×Dic6—22no—
183C2×Dic6—22no—
184C2×Dic6—21no—
185C2×Dic6—24no—
186C2×Dic6—42no—
187C2×Dic6—42no—
188C2×Dic6C22×C412yes—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8#4114yes—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4C22×C412yes—
197C3×C42—24no—
198C3×C42—42no—
199C3×C42C22×C412yes—
200C3×C42C2×D412yes—
201(C2×D3)⋊C4—24no—
202(C2×D3)⋊C4—24no—
203(C2×D3)⋊C4—44no—
204C4⋊Dic3—21no—
205C4⋊Dic3—24no—
206C4⋊Dic3—22no—
207C4⋊Dic3—22no—
208C4⋊Dic3C22×C412yes—
209Dic3⋊C4—24no—
210Dic3⋊C4—24no—
211Dic3⋊C4—24no—
212Dic3⋊C4—24no—
213Dic3⋊C4—24no—
214Dic3⋊C4—44no—
215Dic3⋊C4—44no—
216C4×Dic3—24no—
217C4×Dic3—42no—
218C4×Dic3C22⋊C414yes—
219C4×Dic3C22⋊C414yes—
220C4×Dic3C22×C412yes—
221C4×Dic3C2×D412yes—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8#4114yes—
225ΓL(1,𝔽25)—216no—
226C22×Q8—31no—
227C22×Q8—61no—
228C22×Q8C22×D311yescharacteristic
229C23×C4—31no—
230C23×C4—31no—
231C23×C4—31no—
232C2×M16—64no—
233C2×M16—64no—
234C2×M16—64no—
235C22×C8—32no—
236C22×C8C22×D312yes—
237C4⋊Q8—34no—
238C4⋊Q8—34no—
239C4⋊Q8—62no—
240C4⋊Q8—62no—
241#125—68no—
242C22⋊Q8—34no—
243C22⋊Q8—34no—
244C22⋊Q8—62no—
245C22⋊Q8—68no—
246C22⋊Q8—62no—
247C4×Q8—22no—
248C4×Q8—34no—
249C4×Q8—68no—
250C4×Q8—62no—
251C4×Q8C4×D314yes—
252#117—68no—
253#117—124no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—32no—
257C2×C4⋊C4—61no—
258C2×C4⋊C4—61no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4C22×D312yes—
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C22⋊C4—61no—
268C2×C22⋊C4—61no—
269C2×C22⋊C4—62no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—31no—
273C2×C42—31no—
274C2×C42—31no—
275C2×C42—34no—
276C2×C42—34no—
277C2×C42—61no—
278C2×C42—61no—
279C2×C42C22×D311yescharacteristic
280#103—24no—
281#103—64no—
282#103—64no—
283#103—64no—
284#97—22no—
285#97—62no—
286C4×C8—64no—
287#95—21no—
288#95—62no—
289#95—61no—
290C23×C3C22×D411yescharacteristic
291C22×D3—22no—
292C22×D3—21no—
293C22×D3—22no—
294C22×D3—21no—
295C22×D3C2×C22⋊C412yes—
296C3×Q8—22no—
297C3×Q8—22no—
298C3×Q8—42no—
299C2×C3×C4—21no—
300C2×C3×C4—22no—
301C2×C3×C4—21no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—42no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4C2×C22⋊C412yes—
311C2×C3×C4C2×C22⋊C412yes—
312C2×C3×C4C23×C411yescharacteristic
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C2×#4112yes—
315C2×C3×C4C2×#4112yes—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—42no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3C2×C22⋊C412yes—
326C2×Dic3C2×C22⋊C412yes—
327C2×Dic3C2×C22⋊C412yes—
328C2×Dic3AGL(1,ℤ/8)12yes—
329C2×Dic3#13712yes—
330C2×Dic3#13714yes—
331C4×D3—22no—
332C4×D3—22no—
333C4×D3—24no—
334C4×D3—28no—
335C4×D3—28no—
336Dic6—22no—
337Dic6—22no—
338Dic6—42no—
339Dic6—48no—
340C3×C8—24no—
341C3⋊C8—24no—
342C24—31no—
343C2×Q8—21no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—32no—
347C2×Q8—64no—
348C2×Q8—61no—
349C2×Q8—62no—
350C2×Q8—62no—
351C2×Q8—122no—
352C2×Q8—122no—
353C2×Q8C2×C4×D312yes—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—31no—
358C22×C4—31no—
359C22×C4—31no—
360C22×C4—34no—
361C22×C4—34no—
362C22×C4—32no—
363C22×C4—32no—
364C22×C4—62no—
365C22×C4—64no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—61no—
370C22×C4—61no—
371C22×C4—62no—
372C22×C4D3×D411yescharacteristic
373C22×C4D3×D411yescharacteristic
374C22×C4C23×D311yescharacteristic
375M16—616no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—34no—
379C2×C8—62no—
380C2×C8—62no—
381C2×C8#23514yes—
382C4⋊C4—24no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—32no—
387C4⋊C4—64no—
388C4⋊C4—64no—
389C4⋊C4—64no—
390C4⋊C4—64no—
391C4⋊C4—64no—
392C4⋊C4—64no—
393C4⋊C4—61no—
394C4⋊C4—62no—
395C4⋊C4—62no—
396C4⋊C4—124no—
397C4⋊C4—124no—
398C4⋊C4C2×C4×D312yes—
399C22⋊C4—64no—
400C22⋊C4—64no—
401C22⋊C4—124no—
402C42—24no—
403C42—34no—
404C42—34no—
405C42—32no—
406C42—32no—
407C42—42no—
408C42—64no—
409C42—122no—
410C42C2×C4×D312yes—
411C42D3×D412yes—
412C22×C3C22×C22⋊C411yescharacteristic
413C22×C3#53212yes—
414C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
415C22×C3C2×#13712yes—
416C22×C3C2×#13711yescharacteristic
417C2×D3—22no—
418C2×D3—22no—
419C2×D3—22no—
420C2×D3—42no—
421C2×D3—42no—
422C3×C4—22no—
423C3×C4—22no—
424C3×C4—42no—
425C3×C4—42no—
426C3×C4—82no—
427C3×C4C4×#4111yescharacteristic
428C3×C4C4×#4111yescharacteristic
429C3×C4#52512yes—
430Dic3—22no—
431Dic3—42no—
432Dic3—42no—
433Dic3—82no—
434Dic3#41714yes—
435Dic3#41914yes—
436C23—32no—
437C23—61no—
438C23—61no—
439C23—62no—
440C23—62no—
441C23C2×D3×D411yescharacteristic
442Q8—22no—
443Q8—22no—
444Q8—42no—
445Q8—62no—
446Q8—62no—
447Q8—128no—
448Q8—122no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—32no—
455C2×C4—32no—
456C2×C4—32no—
457C2×C4—34no—
458C2×C4—32no—
459C2×C4—32no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—68no—
467C2×C4—68no—
468C2×C4—62no—
469C2×C4—62no—
470C2×C4—62no—
471C2×C4—64no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4D3×C22⋊C412yes—
481C2×C4D3×C22⋊C412yes—
482C2×C4C22×C4×D311yescharacteristic
483C2×C4C2×#23512yes—
484C2×C4C2×D3×D411yescharacteristic
485C2×C4D3×#4112yes—
486C8—24no—
487C8—64no—
488C2×C3#229711yescharacteristic
489C2×C3C2×#41911yescharacteristic
490C2×C3C2×#41711yescharacteristic
491C2×C3#318212yes—
492C2×C3#318512yes—
493D3—42no—
494C22—62no—
495C22—62no—
496C22—62no—
497C22—122no—
498C22—122no—
499C22C2×D3×C22⋊C411yescharacteristic
500C22#554812yes—
501C22D3×AGL(1,ℤ/8)11yescharacteristic
502C22D3×#13711yescharacteristic
503C22D3×#13712yes—
504C4—22no—
505C4—22no—
506C4—34no—
507C4—34no—
508C4—42no—
509C4—42no—
510C4—62no—
511C4—82no—
512C4—122no—
513C4—122no—
514C4—242no—
515C4C4×#23511yescharacteristic
516C4#552211yescharacteristic
517C4#554712yes—
518C3#2140911yescharacteristic, complemented, Sylow
519C2—122no—
520C2#7356311yescharacteristic
521C2D3×#41711yescharacteristic
522C2D3×#41911yescharacteristic
523C2#7979212yes—
524C2#7983812yes—
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1763C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
cosocleC22×C3×C4C24
hypercenter#1763D3
hypocenterC3#21409
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C3×#1763C2semidirect
5. #71631C2semidirect
7. #70225C2semidirect
8. #69813C2semidirect
9. #69475C2semidirect
11. #68803C2semidirect
18. #5578C4semidirect
57. #1763D3semidirect
518. C3#21409semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#55782
C2×#47072
#737532
C3×#17632
#716312
#702712
#702252
#698132
#694752
#688842
#688032
#214093